The case for black hole thermodynamics Part I: phenomenological thermodynamics

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1 The case for black hole thermodynamics Part I: phenomenological thermodynamics David Wallace October 7, 2017 Abstract I give a fairly systematic and thorough presentation of the case for regarding black holes as thermodynamic systems in the fullest sense, aimed at students and non-specialists and not presuming advanced knowledge of quantum gravity. I pay particular attention to (i) the availability in classical black hole thermodynamics of a well-defined notion of adiabatic intervention; (ii) the power of the membrane paradigm to make black hole thermodynamics precise and to extend it to local-equilibrium contexts; (iii) the central role of Hawking radiation in permitting black holes to be in thermal contact with one another; (iv) the wide range of routes by which Hawking radiation can be derived and its back-reaction on the black hole calculated; (v) the interpretation of Hawking radiation close to the black hole as a gravitationally bound thermal atmosphere. In an appendix I discuss recent criticisms of black hole thermodynamics by Dougherty and Callender. This paper confines its attention to the thermodynamics of black holes; a sequel will consider their statistical mechanics. 1 Introduction Black hole thermodynamics (BHT) is perhaps the most striking and unexpected development in the theoretical physics of the last forty years. It combines the three main areas of fundamental theoretical physics quantum theory, general relativity, and thermal physics and it offers a conceptual testing ground for quantum gravity that might be the nearest that field has to experimental evidence. Yet BHT itself relies almost entirely on theoretical arguments, and its most celebrated result Hawking s argument that black holes emit radiation has no direct empirical support and little prospect of getting any. So to outsiders to physicists in other disciplines, or to philosophers of science the community s confidence in BHT can seem surprising, or even suspicious. Can we really be so confident of anything without any grounding in observation? Dornsife College of Letters, Arts and Sciences, University of Southern California; dmwallac@usc.edu 1

2 In this article, and its sequel, I want to lay out as carefully and thoroughly as I can the theoretical evidence for BHT. It is written with the zeal of the convert: I began this project sharing at least some of the outsiders scepticism, and became persuaded that the evidence is enormously strong both that black holes are thermodynamical systems in the fullest sense of the word, and that their thermodynamic behaviour has a statistical-mechanical underpinning in quantum gravity (and, as a consequence, that black hole evaporation is a unitary process not different in kind from the cooling of other hot systems, and that it involves no fundamental loss of information). There are of course many reviews of this material. But those I know either (i) take for granted the main results of BHT, moving quickly over established material to get students up to speed with the research frontier; (ii) are explicitly historical, which illuminates how the community in fact came to accept BHT but can obscure the logic of whether and why they should have accepted it, or (iii) are written at a very high level of mathematical rigor, so high that a large fraction of the literature has to be omitted. I hope this paper will be complementary to extant material. With few exceptions, I present and describe results without going into the details of their derivation, and the student who wishes to properly understand the topic will need to read this paper in parallel with some of the extant review literature. My starting points (for this part of the paper) were Harlow (2016), Jacobson (1996, 2005), Thorne, Price, and Macdonald (1986), and Wald (1994, 2001). A note on mathematical rigor: the tendency in foundational work on this subject (see, e. g., Belot, Earman, and Ruetsche (1999) and Earman (2011)) has been to work at the level of rigor typical in mathematical physics, where all results are stated exactly and proved rigorously. This is much higher than the standard in theoretical physics more generally; it has the advantage of reliability, but the disadvantage that a very large fraction of the literature must be elided especially in a frontier area like this, where the underlying physical principles are unclear and the mathematical framework partial and under active development. And the case for BHT as will become apparent throughout this paper and, even more so, its sequel rests not so much on individual results that have been established with full precision and rigor, but on the many independent calculations with different premises and approximation schemes that all lead to the same result. So this paper is written at the theoretical-physics level; I hope that readers who prefer their mathematics more precise will at least get a sense as to why the community takes BHT so seriously, even if they are not persuaded themselves. This is a large topic, too large for any one paper. In this paper I confine my attention to phenomenological thermodynamics, setting aside any considerations of statistical-mechanical underpinnings for that thermodynamics. In Wallace (2017a) I consider the progress made in calculating the thermodynamical properties of black holes via statistical mechanics (in effective-field theory quantum gravity, in string theory, and via the AdS/CFT correspondence). And in Wallace (2017c) I use these two papers as a starting point to review and assess the notorious information-loss paradox which has motivated a large part 2

3 of the critical attention paid to BHT. The structure of the paper is as follows. I begin in section 2 by briefly reviewing classical thermodynamics, and discussing how it is modified for selfgravitating systems: to see whether black holes are thermodynamical, we need to be clear what thermodynamics is in the first place. In section 3 I consider classical black hole thermodynamics, arguing that while black holes offer a strikingly good realisation of the principles of thermodynamics when regarded as isolated systems, they completely fail to do so when considered as components of a larger system. In section 4 I show how including the implications of quantum field theory, in particular (though not exclusively) the Hawking effect, entirely remove this limitation; I also review the strength of the evidence for the Hawking effect itself, and the related but logically stronger claim that Hawking radiation leads to black hole evaporation. In an appendix, I address the arguments of a recent paper by Dougherty and Callender (2016) which criticises BHT (that paper was one trigger for my writing this paper, but engaging with its arguments in the main text would complicate my structure unhelpfully). I assume some familiarity with classical general relativity (in particular the Schwarzschild solution) and classical thermodynamics (and I quote standard results from both fields without explicit references); a little prior exposure to quantum field theory would also be helpful in section 4. Except where explicitly noted, I adopt units where G = = c = k B = 1. 2 Thermodynamics and statistical mechanics: a brief review Without any pretension to historical accuracy, complete precision or logical independence, we can break the salient parts of equilibrium thermodynamics into three: equilibrium and equilibration; the First and Second Laws for individual systems; interactions between multiple systems. I discuss each in turn; I then briefly consider the generalisation of equilibrium thermodynamics to local thermal equilibrium, and the subtleties introduced by gravitation. For this paper I do not need, and do not discuss, the statistical-mechanical underpinnings of thermodynamics. 2.1 Equilibrium and equilibration A thermodynamic system has a family of equilibrium states parametrised by the energy and by a (usually small) number of additional conserved quantities and/or external constraints. If the system is in the equilibrium state corresponding to its constraints and conserved quantities, it remains in that state; if it is not, it equilibrates, evolving towards that state and reaching it, to any given degree of accuracy, after a finite time (Brown and Uffink (2001) refer to this equilibration principle as the Minus First Law of Thermodynamics). The work done by such a process is defined as the change in the system s energy, 3

4 and (by conservation of energy) is then equal to the energy cost to the external agent. For instance, for a box of gas (of some fixed kind of particle) the external constraint is the volume of the box, and the conserved quantities are the energy, the number of particles, and in principle the momentum and angular momentum. In general we assume a nonrotating box and study it in its rest frame, and/or assume that the box is so massive not to be affected by particle collisions, so that momentum and angular momentum may be neglected and energy and internal energy can be identified; often we also take the particle number as fixed and do not include it explicitly as a variable. 2.2 The First and Second Laws for individual systems Given an isolated thermodynamic system, an adiabatic transformation of that system is some operation performed on the system, starting at equilibrium, that transforms its state to another equilibrium state without coupling it nontrivially to other thermodynamics systems. Any such transformation can be thought of as a change to the external constraints and conserved quantities of the system via some external force; paradigm examples include expanding or compressing a gas, or putting a non-rotating system into rotation. Only some such changes are physically possible by means of adiabatic transformations. Specifically, if the system s equilibrium states are parameterised by energy U and conserved quantities/external constraints X i, there exists a function S(U, X 1,... X N ), called the entropy of the system (and hence defined, as far as thermodynamics is concerned, only at equilibrium), such that S is non-decreasing under any adiabatic transformation. This entropy non-decrease law is one form of the Second Law of Thermodynamics. Adiabatic transformations can then be broken into three categories: reversible transformations, which leave S unchanged; irreversible transformations, which increase S, and thermodynamically forbidden transformations, which decrease S. It is generally the case that all reversible and irreversible transformations are physically performable (at least in principle, and perhaps in an idealised limiting case) so that the Second Law imposes a necessary and sufficient condition for a transformation to be possible. In particular, if we make a very small adiabatic change to the X i and then wait for the system to re-equilibrate, that change will leave S unchanged to a very high degree of accuracy. So sufficiently slow adiabatic changes to the X i will define processes which are very close to being reversible, becoming exactly reversible in the infinite-time limit. It is generally the case that such quasi-static transformations are always available. We can express the entropy in differential form as ( ds = β du + ) λ i X i (1) i 4

5 or, rearranging so that U is a function of S and the X i, du = T ds i λ i X i (2) where T = 1/β. T is called the thermodynamic temperature and the λ i are the thermodynamic variables conjugate to the X i ; they can be given explicitly by ( ) ( ) 1 S S T = ; λ i = T. (3) U X i X i U,X j or by T = ( ) ( ) U U ; λ i =. (4) S X i X i S,X j The λ i usually have a physical meaning: in particular, the variables conjugate to volume, momentum, angular momentum, particle number, and charge are, respectively, pressure, centre-of-mass velocity, angular velocity, chemical potential, and electric potential. Equation (2) is one form of the First Law of Thermodynamics. It can be understood entirely statically, as a statement of the relations between different equilibrium states. But given the existence of quasi-static processes, we can also interpret it as describing the actual change in U induced by small adiabatic changes X i X i + δx i to the parameters, together with a flow of energy Q = T δs into the system from some external reservoir. Following Wald (1994, p.141)) we can call these the equilibrium-state and physical-process interpretations, respectively. Flow of energy of this kind is called heat flow and makes sense even if the flow is not infinitesimal; conservation of energy entails that the change in a system s energy equals the heat flow into it plus the work done on it, which is another form of the First Law. Finally, note that at this stage of our analysis S (and, hence, T ) is fixed only up to an arbitrary rescaling: we can replace S with f(s), for any smoothly increasing function f, and 1/T with f (1/T ), without affecting anything said so far. 2.3 Multiple thermodynamic systems Much of the content of thermodynamics is only available once we allow dynamical interactions between multiple systems. The rules for doing so are: 1. Any two systems may be placed in thermal contact, so that heat may flow between them while their other conserved quantities and external parameters remain separately fixed. This can be generalised to allow for other kinds of contact in which the two systems can exchange other conserved quantities. 2. Multiple systems in (perhaps-generalised) thermal contact may be treated as a single system; in particular, any such combined system will have a unique equilibrium state. 5

6 3. The Second Law of Thermodynamics generalises to require that the total entropy of two systems in (perhaps-generalised) thermal contact does not decrease when those systems exchange energy and other conserved quantities. For this to be well-defined, the possibility for rescaling of entropy decreases sharply: in multiple-system contexts, entropy must be taken as fixed up to a system-independent scale and a system-dependent additive constant. From (2) and (3) together, it follows that: 4. If two systems are in thermal contact, and heat δq flows from system 1 to system 2, the total change in entropy is δs = δq(1/t 2 1/T 1 ). So heat will flow only if T 1 > T 2, and indeed, no process can as its sole effect induce heat flow unless this condition holds (the Clausius statement of the Second Law). It follows that a necessary and sufficient condition for two systems in thermal contact to be jointly at equilibrium is that they are separately at equilibrium with equal temperatures. (This generalises to other forms of contact.) As a consequence, the relation at equilibrium with is an equivalence relation: this is the Zeroth Law of thermodynamics, and in textbook presentations is often taken as a starting point; in my presentation, it is a consequence of other assumptions. 5. Given a process involving an infinitesimal heat flow between two equilibrium systems at thermodynamic temperatures T 1, T 2 together with work W done on the combined system, and such that the conserved quantities and external constraints of the two systems (other than energy) are unchanged at the end of the process, the First Law entails that ( W = T 1 S 1 + T 2 S 2 = T 1 S 1 + T ) 2 S 2. (5) T 1 Since the Second Law entails that S 2 S 1, we have W T 1 S 1 (1 (T 2 /T 1 )). (6) From this, we can read off that the maximum efficiency of any cyclical process which generates work from heat flow between the two systems is (1 T 2 /T 1 ) and, a fortiori, that no cyclical process can as its sole effect convert heat into work, which is the Kelvin statement of the Second Law. (Other processes can do better, but they do not leave the other conserved quantities and constraints unchanged and so cannot be performed in a cycle.) 2.4 Local thermodynamic equilibrium In an extended body (such as a solid, a fluid, or a field), if the rate at which a small region of the fluid equilibrates is fast compared to the rate at which it exchanges energy and other conserved quantities with neighboring regions, the 6

7 body will approach local thermal equilibrium, at which we may express thermodynamic quantities like charge, energy, entropy, temperature and pressure as functions of position in the body. (For instance the sun, while not at equilibrium, is at local equilibrium, so that we can describe how temperature, pressure, entropy density and energy density vary from the core to the atmosphere.) Various phenomenological equations can be derived or postulated to describe the flow of thermodynamic quantities through the system. For instance, Ohm s Law describes how current flow in a conductor is dissipated as heat, and the Navier-Stokes equations describe the flow of a viscous fluid and the dissipation of organised energy as heat in that fluid. Various transport coefficients, like electrical resistivity and viscocity, appear in those equations, so that they cannot simply be derived from the equation of state but require additional empirical input. 2.5 Complications of gravity Thermodynamics can be coherently formulated for (relativistic or Newtonian) self-gravitating systems, but the existence of long-range forces in these systems leads to important subtleties, even before we consider black holes. Rather than discuss the (somewhat controversial) general structure of these subtleties (for that discussion, see Wallace (2010), Callender (2011), and references therein), I will illustrate them with a concrete example due to Sorkin, Wald, and Jiu (1981): a spherical box of radiation at thermal equilibrium, potentially large enough that self-gravitation has a discernible effect. The sphere is assumed to be nonrotating and at rest. Its equation of state depends on two parameters: its radius R and its mass M (for a relativistic system, and in units where c = 1, its mass and its energy can be identified). A crucial parameter is the Schwarzschild radius R S (M) = 2GM/R: if R < R S (M) then an event horizon forms around the sphere and it must be treated as a black hole. As long as R R S (M), gravitating effects are fairly insignificant and the sphere may be treated as if it were non-self-gravitating. It then behaves as a pretty conventional thermodynamic system, with an extensive equation of state determined by the intensive formulae ρ = bt 4 ; s = 4 3 bt 3 (7) that determine the energy density ρ and entropy density s as functions of the temperature. In particular, if we consider a sequence of successively larger spheres with M/R 3 held constant, the temperature and density of each sphere likewise remain constant. But for denser spheres (the transition occurs roughly around R 5R S ) gravitational effects become highly important and the system displays several distinctive features characteristic of strongly self-gravitating systems (all discussed, or readily derived, in Sorkin et al s paper): 1. Because spacetime is nontrivially curved within the sphere, we cannot define the mass of the sphere simply as the integral of the local mass- 7

8 density: indeed, that integral is not even well-defined in a coordinate-free way. Instead, the mass can defined by using Noether s theorem (according to which energy is the conserved quantity associated with time translation symmetry), calculated at a distance much larger than the shell radius at which the spacetime is approximately flat. The precise version of this concept of mass is called the ADM mass, after Arnowitt, Deser, and Misner (1962) (a related version, the Bondi-Sachs mass (Bondi 1960, Sachs 1961, 1962), is better suited to handle situations involving radiation but rests on the same basic idea). If the sphere had non-trivial spatial momentum and/or angular momentum, analogous ADM momenta and angular momenta can also be defined, using the appropriate asymptotic Noether symmetries. 2. The sphere becomes increasingly non-homogeneous, with the density being much higher towards the centre of the sphere. From this and the local equation of state (7), we can deduce that the locally-measured temperature also increases closer to the centre. The locally measured temperature t(r) at a radius r from the centre is related to the thermodynamic temperature (given by 1/T = S/ U) by t(r) = α(m(r), r) 1 T (8) where m(r) is the mass of the sphere internal to r (more precisely: the ADM mass that the region of the sphere interior to r would have if it were confined to that region and the rest of the sphere removed) and α(m, r) = (1 2Gm/r) 1/2 is the gravitational redshift induced by a spherically symmetric mass m. 3. The sphere is no longer extensive in any meaningful sense: increasing R to KR and M to K 3 M will not produce a qualitatively similar sphere. Indeed, if R < 0.254R S, the sphere becomes unstable and undergoes gravitational collapse into a black hole. 4. The heat capacity of the sphere (i. e., the rate of change of mass with temperature at constant radius) decreases to zero and becomes negative, so that decreasing the energy of the sphere actually causes it to become hotter. Though Sorkin et al do not discuss it, the notion of thermal contact also has to be analysed with some care for these systems. For a start, we cannot put two such spheres in thermal contact simply by placing them adjacent to one another: their mutual gravitation would radically alter each other s states, probably producing gravitational collapse unless handled carefully. An intermediate system is required. As a concrete example, consider the following process for transferring heat between two spheres with thermodynamic temperatures T 1, T 2, masses M 1, M 2 and surface redshifts α 1, α 2 : 8

9 1. A box is slowly lowered to the surface of Sphere 1 from infinity (i. e., from very far above the sphere), allowed to fill with a small amount of radiation of local mass m and temperature T 1 /α 1, and then slowly lifted back to infinity, requiring (Unruh and Wald 1982) work W 1 = (1 α 1 )m. (9) 2. The box is adiabatically compressed or expanded (as appropriate) to a temperature T 2 /α 2, requiring additional (possibly negative) work ( ) (T2 /α 2 ) W 2 = (T 1 /α 1 ) 1 m (10) (as can be deduced from the equation of state (7)) and changing its mass to m(t 2 /α 2 )(T 1 /α 1 ) 3. The box is slowly lowered to the surface of Sphere 2, requiring negative work W 3 = (1 α 2 ) (T 2/α 2 ) m. (11) (T 1 /α 1 ) 4. The box is then opened and the radiation released into Sphere 2; this is adiabatic, since it has the same local temperature as Sphere 2 s surface. The entire process is adiabatic and has the following energy implications: M 1 = α 1 m; M 2 = α 1 m(1+(t 2 /T 1 )); W = W 1 +W 2 +W 3 = α 1 m(t 2 /T 1 ). (12) This has the characteristic form of a Carnot cycle. As a corollary, if T 1 > T 2, net work is extracted by the process, and we can replace (3) by 3. The box is slowly lowered towards the surface of Sphere 2 until the work extracted by doing so makes the whole process work-neutral, and then released to free-fall the rest of the way. The new process permits heat transfer, without work expenditure, from Sphere 1 to Sphere 2 provided T 1 > T 2, and so provides a means to put the two spheres in (somewhat indirect) thermal contact. In many examples of self-gravitating bodies, there is another way to put two bodies into thermal contact: seal them both into a very large box with reflecting walls, and wait. If one or other body is above absolute zero, it will emit electromagnetic radiation; in due course, the box will fill with radiation in local thermal equilibrium. Each body is in thermal contact with the radiation and so, indirectly, with the other body. This is an effective way (in principle and in thought, not in engineering practice!) to, for instance, place two neutron stars or white dwarfs into thermal contact. It is not really an option for our radiation spheres, because they are themselves comprised of thermal radiation so the breakdown into subsystems would not be well-defined. 9

10 3 Classical black hole thermodynamics We can now consider whether, and to what extent, these thermodynamic notions apply to black holes and systems of black holes. In this section I consider only classical black holes, by which I mean: black holes, if we neglect or imagine away any quantum-field-theoretic effects: in particular, any matter fields present will be treated phenomenologically and classically. For clarity, I do not mean black holes, under the fiction that the world is exactly classical : I m not sure that is even well-defined (though see Curiel (2014)) but in any case it presumably would not include thermal radiation, which can be treated phenomenologically as a classical fluid but whose derivation via statistical mechanics requires quantum theory. 3.1 Black holes as objects The basic idea of BHT is that black holes are thermodynamic systems, and that a particular subclass of black holes (the stationary black holes) are the equilibrium states of those systems. But from the starting point of general relativity, it is hard to see how this is even coherent: in that context, a black hole is identified globally as a region of spacetime from which null geodesics cannot reach future infinity (see, e. g., Hawking and Ellis (1973)). A spacetime region cannot itself change in time, so the notions of equilibrium or equilibration don t obviously make sense under this definition. But the relativist s concept of a black hole is not the only one extant in physics. Astrophysicists have long spoken of black holes as objects which persist through time and whose properties change in time: any talk of black holes orbiting one another, or of two black holes merging to form a larger hole, or of the velocity of a black hole relative to another astrophysical object, seems to require a three-dimensional view of black holes as objects, in tension with the spacetime-region view natural in theoretical relativity. The membrane paradigm of Macdonald, Price and Thorne, developed in detail in the astrophysical context in Thorne, Price, and Macdonald (1986) and adapted for the quantum theory of black holes by Susskind, Thorlacius, and Uglum (1993), addresses just this problem. Thorne et al consider a timelike surface the membrane, or stretched horizon that is placed around the true event horizon, at a very small proper distance from the true horizon. Thorne et al give the stretched horizon an area (1 + α) 2 times that of the true horizon, where α is some positive real number 1; more useful for foundational purposes is Susskind et al s convention (which I adopt henceforth), giving the horizon an area one Planck area larger than that of the true horizon. The defining property of the event horizon, physically, is that nothing can emerge from it, and so in particular nothing can enter it and later return. But virtually nothing can cross the stretched horizon and return, because doing so would require extremely high accelerations indeed, under Susskind et al s convention, it would require accelerations so high as to require Planck-scale physics to describe. So as long as we are dealing with energy levels below the 10

11 Planck scale, the stretched horizon may be treated as a one-way barrier just as can the true horizon. On the other hand, the stretched horizon is an ordinary timelike surface; it can be treated as a two-dimensional closed surface in space that evolves through time, and so can be attributed potentially-time-dependent physical properties. And with its aid, we can then restate the goal of black hole thermodynamics as follows: to investigate the extent to which the stretched horizons of black holes can be treated as ordinary physical systems, and assigned mechanical, electromagnetic, and thermodynamic properties, from the point of view of any observers who remain outside the black hole or, to put it in less operational terms, the extent to which we can give a self-contained account of physics in the region of spacetime exterior to any black holes in terms of stretched horizons to which such properties are assigned. 3.2 Equilibrium and equilibration for black holes Thermodynamics describes equilibrium systems in terms of their conserved quantities and external constraints. There are no real external constraints applicable to a black hole, but there are quantities which we would expect to be conserved: the energy, momentum and angular momentum of the hole (defined asymptotically by the ADM method) and its electrical charge. In each case these quantities are associated to long-range forces (gravity for the quantities associated to spacetime symmetries; electromagnetism for charge), as these forces ensure that matter bearing the conserved quantity will leave an asymptotic trace on the spacetime even once it crosses the stretched horizon. (Conserved quantities like baryon number, by contrast, cannot be expected to show up in the physics of the black hole exterior, since the long-range physics will be indifferent as to whether a particle that crosses the horizon is, say, a neutron rather than an anti-neutron.) By working in a reference frame at which the black hole is at rest and its angular momentum is aligned along the z axis (again, using the ADM charges to define this rigorously) we reduce the conserved quantities to three: the black hole s mass M, the magnitude J of its angular momentum, and its charge Q. So if black holes have equilibrium states, we would expect the space of such states to be parametrised by these three quantities. The definition of an equilibrium state is that it is unchanging in time, and general relativity offers a clear way to represent this: we look for stationary solutions of the Einstein field equations, that is: solutions with a timelike Killing vector. Such solutions certainly exist for general M, J, Q: the Kerr-Newman solutions to the coupled equations of general relativity and vacuum electromagnetism (aka Einstein-Maxwell theory) are stationary and parametrised precisely by mass, angular momentum and charge. When Q=0, these solutions reduce to the Kerr solutions of vacuum general relativity; when J = 0, to the sphericallysymmetric Reissner-Nordstrom solutions of the Einstein-Maxwell theory; when both are zero, to the well-known Schwarzschild solution. The Kerr-Newman solution only describes a black hole when Q 2 + J 2 /M 2 M 2, with solutions violating this inequality describing naked singularities; black holes that saturate 11

12 the inequality are called extremal, and are a somewhat puzzling but theoretically important special case. The 1970s saw extensive work by Bardeen, Carter, Hawking, Israel and many others to prove the No-Hair Conjecture : that the Kerr-Newman black holes are the unique stationary solutions to the Einstein-Maxwell theory, and so provide unique equilibria. To this day there remain loose ends in the conjecture and in its extension to more general situations in higher spacetime dimensions and with other long-range forces present, but in his review article in the Einstein Centenary Survey (Carter 1979) felt able to say that the no-hair theorems available... are quite sufficient to justify with at least the degree of rigour usually considered acceptable in physics the assumption by any practically minded astrophysical theorist that any (external source free) black hole equilibrium-state solution... belongs to the Kerr or Kerr-Newman families. (See Carter s review article for detailed references and for a summary of the main results; see also Carter (1997) for some historical remarks and Chrusciel and Costa (2008) for a fairly up-to-date survey.) Of course, thermodynamic equilibrium requires more than mere stationarity: it requires non-equilibrium systems to converge to equilibrium, and in particular, perturbations of equilibrium states to be damped back down to equilibrium. The stability of black holes, and the convergence to equilibrium of non-stationary black holes, has been extensively studied both analytically and numerically. By the mid-1980s (see chapters VI-VII of Thorne, Price, and Macdonald (1986), and references therein) it was established that perturbations of the stretched horizon by external gravitating bodies are damped away (for instance, the stretched horizon can oscillate, but these oscillations are damped, dying away back to equilibrium via the emission of gravity waves). Computer simulations of colliding black holes, and accretion of matter onto black holes, likewise demonstrate that the system evolves rapidly to the equilibrium-blackhole configuration, decaying by the emission of gravity waves ( ringdown ). And the historic observation of gravity waves in 2016 by the LIGO observatory provided a remarkably precise fit to the quantitative ringdown predictions, and so can reasonably be said to provide (ongoing) observational support for black hole equilibration. In summary: we have both a clear understanding of what the black hole equilibria are, and a pretty good grasp on why they are indeed equilibria: at the least, I think it would be hard to argue that we have any better theoretical control of how paradigm normal thermodynamical systems, like dilute gases, approach and remain at equilibrium. So far, black holes fully fit the requirements to count as thermodynamic systems. 3.3 The laws of black hole thermodynamics To treat a black hole as a thermodynamic system requires us to identify external interventions, and to divide them into adiabatic changes and heat flows. The 12

13 former is fairly straightforward: to move a black hole from one equilibrium state to another is going to require us to change its mass, angular momentum or charge, and the simplest way to do that is to drop matter into it. The latter is more delicate, since the division between heat and work is less obvious in an alien situation like this than for a box of gas. The simplest thing to do (in this case as in other less-familiar cases in regular thermodynamics) is to identify which transformations are reversible and which irreversible, and then define the quasi-static adiabatic processes as the reversible ones. Christodolou and Ruffini demonstrated (Christodolou and Ruffini (1971); see Misner, Thorne, and Wheeler (1973, pp ) for a discussion) that the quantity that plays the role of entropy for a black hole (at least for infinitesimal changes) is surface area (which, for an equilibrium black hole, is given by a known function of M, J and Q): any intervention on an equilibrium black hole must leave the surface area nondecreasing, so that the reversible processes are those that leave surface area invariant and the irreversible processes strictly increase area. Reversible transformations of J and Q can be brought about as follows: To reversibly change the charge of a charged black hole, lower some charged matter very slowly on a cord so that it is suspended, stationary, just above the event horizon; then let go. To reversibly increase the angular momentum of a rotating black hole, fire some mass at it on a trajectory which just brushes the event horizon. To reversibly decrease the angular momentum of a rotating black hole, use the Penrose process (Penrose (1969), Penrose and Floyd (1971); see Carroll (2003, pp ) for an introduction): fall freely towards the black hole on a trajectory that passes just above the event horizon, and at point of closest approach, eject some mass into the black hole on a trajectory opposite to the direction of rotation of the hole. Dropping charge into a black hole from finite height, or injecting mass on a non-brushing trajectory, or using the Penrose process on a higher trajectory, will in each case be irreversible, bringing about an increase in surface area. Hawking s area theorem (Hawking 1972) generalises Cristodolou and Ruffini s result beyond infinitesimal changes: Hawking proved that the area of any black hole is nondecreasing. His derivation presumes 1. that physics in the exterior of the black hole remains predictable (that is, roughly: assuming that no naked singularities form; see Wald (1994, pp.138-9) for a more precise discussion); 2. the Einstein field equations; 3. the null energy condition: that the stress-energy tensor T satisfies T (v, v) 0 for any null v. This is violated in some exotic quantum-field-theoretic situations (of which more later) but seems a safe assumption for bulk matter, such as electromagnetic radiation and astrophysical fluids. 13

14 Bardeen, Carter, and Hawking (1973) christened the Area Theorem the Second Law of black hole thermodynamics ; in fact, it goes rather beyond the entropy-increase form of the standard Second Law, since black hole surface area remains well-defined even when a black hole is far from equilibrium, whereas thermodynamic entropy is defined only at equilibrium. In the same paper, Bardeen et al also established the First Law of black hole thermodynamics which states that dm = 1 κda ΩdJ ΦdQ (13) 8π where κ is the surface gravity of the black hole, A its surface area, Ω its angular velocity, and Φ the electric potential on its surface. This is precisely the form of the standard First Law for a thermodynamic system where angular momentum and charge are conserved quantities, including the identification of the conjugates to J and Q as, respectively, angular velocity and electric potential. It permits us to identify the thermodynamic temperature of the hole as proportional to the surface gravity albeit, as long as we are considering a system in isolation, we have only identified entropy up to a monotonic function. Furthermore, we can independently prove the physical-process and equilibrium-state versions of the First Law distinguished by Wald (recall the discussion in section 2.2), demonstrating that the overall structure of interventions on the black hole is self-consistent and fits the model of equilibrium thermodynamics. 3.4 Beyond Einstein s equation Bardeen et al s derivation of the laws of Black Hole thermodynamics presupposed the Einstein field equations; however, as Wald and collaborators have shown (Wald (1993); see Wald (1994, pp ) for an introduction and further references, and Jacobson and Mohd (2015) for more recent developments), the First Law (in both physical-process and equilibrium-state form) can be derived from a general diffeomorphism-invariant Lagrangian theory of gravity by identifying the entropy as (a form of) the Noether charge associated with the diffeomorphism symmetry, evaluated with respect to a vector field that coincides on the horizon with the horizon Killing vector. So far as I know there is no fully general non-decrease theorem for this generalised black hole entropy of the same scope of Hawking s area theorem, but Jacobson, Kang, and Myers (1995) have demonstrated that this generalised definition of entropy is nondecreasing under at least quasi-stationary processes, provided that the null energy condition is satisfied; they also prove the analog of Hawking s result for a large class of generalisations of the Einstein Lagrangian. The physical reason for caring about this generalisation lies in the effectivefield-theory program in contemporary particle physics. From that perspective, general relativity is thought of as a non-renormalisable effective field theory, regularised by a cutoff imposed by unknown Planck-level physics. In such a theory, all possible diffeomorphism-covariant action terms should be present; the Einstein-Hilbert action is just the leading-order term in an infinite expansion of 14

15 the Lagrangian in these various terms. So the fact that black hole thermodynamics extends so naturally beyond the Einstein-Hilbert case is reassuring for the physical applicability of the theory. 3.5 Local properties of the stretched horizon The stretched horizon of a black hole is, it seems, a purely fictional entity, invisible to anyone falling through it and corresponding to no locally-present distribution of charge or energy. It is therefore frankly startling that it can be treated not simply as a formal device to make sense of black hole thermodynamics (as I used it above) but as an actual extended physical system with local thermodynamic properties. To expand: as discussed in extenso in Thorne, Price, and Macdonald (1986) and references therein, we can treat the stretched horizon as a two-dimensional, electrically-conducting, viscous fluid, assigning to each infinitesimal part of its surface the exact charge, current, and stress-energy densities required to terminate the electromagnetic and gravitational field lines on its exterior. This assignment is arguably fictional since an observer freely falling through the horizon will not encounter these charges or energies, but from the point of view of physics outside the stretched horizon they are entirely real. To give some examples (many more can be found in Thorne et al): 1. If a positively charged particle falls towards the North pole of an uncharged black hole, its field will induce a current flow of negative charge towards the north pole, which will become negatively charged; the South pole, opposite the direction of approach of the falling particle, will become positively charged. By applying the law of Ohmic dissipation to this current flow (the black hole s surface resistivity is 377 ohms) we deduce that heat will be dissipated in this process so that the black hole area increases. When the charged particle reaches the surface, current will flow back until the charge density on the surface is constant, dissipating more heat. Any region of charge excess will spread out exponentially so that the time for an initially non-equilibrium charge distribution to equilibrate is τ eq = M log M in Planck units, or in more astrophysically useful units τ eq ( M M ) (log(m/m ) ) seconds (14) (M = kg is the mass of the Sun). Only in the limit where the charge is lowered infinitely slowly to the surface will the current flow be so slow, and the readjustment of charge across the surface so complete, that no heat is dissipated; this is the reversible process described previously.(znajek 1978; Damour 1978; Macdonald and Suen 1985; Thorne, Price, and Macdonald 1986, pp.35 38,57 64.) 2. If an electrically neutral black hole rotates in an asymptotically constant magnetic field at right angles to its axis of rotation, eddy currents will 15

16 be induced in the horizon. The magnetic field will exert a torque on the black hole via these currents, which will slow its rotation while also dissipating heat through electrical resistance. The result is that the rotational energy of the black hole will be dissipated as heat, slowing the black hole s rotation and increasing its area; the overall energy of the black hole remains conserved: that is, no energy is extracted from the static magnetic field in this process. (Thorne and Macdonald 1982,(Thorne, Price, and Macdonald 1986, pp ).) 3. If a black hole rotates in the tidal field of a larger gravitating body, the surface of the hole will be perturbed; this in turn produces viscous dissipation and corresponding viscous torque on the black hole in accord with the Navier-Stokes equation, dissipating heat and slowing the rotation of the hole. (Hawking and Hartle 1972; Hartle 1973, 1974; Thorne, Price, and Macdonald 1986, pp ) Also part of the local thermodynamics of black holes is the so-called Zeroth law of black hole thermodynamics (Bardeen, Carter, and Hawking 1973), which states that the temperature of a black hole is constant everywhere on the horizon. In ordinary thermodynamics, the analogous result that for a body at equilibrium, the local temperature is constant is more naturally thought of as a corollary of the Zeroth Law applied to the local-thermal-equilibrium context. 3.6 No thermal contact for classical black holes So far as we treat each black hole as an isolated system, the resemblance to a thermodynamic system seems pretty complete: black holes have notions of equilibrium and equilibration, reversibility and irreversibility, and local thermodynamic properties. But the resemblance terminates abruptly at least as far as classical black holes are concerned as soon as we try to consider them as thermodynamic systems interacting with other black holes, or with non-black-hole thermodynamic systems. Specifically: there seems to be no available process that can reduce the entropy of one black hole and increase that of another (or of a non-black hole thermodynamic system), even if the total entropy is increasing. To the contrary, the analysis of reversible and irreversible processes above applied to each hole separately. Likewise, Hawking s area theorem applies separately to each connected component of a spacetime s event horizon, and so mandates not just that the total entropy of a system of black holes is nondecreasing but that the entropy of each black hole is separately nondecreasing. As a corollary, there seems no prospect of running a Carnot cycle between two black holes, and no prospect of allowing heat to flow from one hole to another. Likewise, there seems no way to make sense of heat flow from a black hole, to any other thermodynamic system. The nearest we can get is to allow two black holes to interact by colliding, in which case the area theorem guarantees that the new black hole has a larger entropy than its constituents, but this is a pale shadow of genuine thermal contact. 16

17 In particular, classical black holes are completely black in the sense that they omit no thermal radiation. This means that a black hole placed in thermal contact with another body by the method of putting both in a box and letting it fill with radiation will simply eat all the radiation, however low its temperature. The only temperature that we seem consistently able to attribute to a classical black hole is then absolute zero. These limitations are aggravated by Bekenstein s (1973) observation that identifying black hole area with entropy also provides opportunities to violate the Second Law of thermodynamics unless we place some constraints on the form of the energy-entropy relation for ordinary matter constraints that do not seem well motivated within classical physics. Specifically: If some body of small mass m and entropy s is slowly lowered right to the event horizon and then released (the so-called Geroch process, proposed by Robert Geroch during a 1970 Princeton colloquium), it will do work on the mechanism that lowers it. Qualitatively this is no different from the way in which a weight slowly lowered from a pulley can do work at the top of the pulley, but the quantitative scale is much larger: if a pointlike body of mass m is slowly lowered to a point above the event horizon with redshift α, then the work extracted is W = m(1 α) and so (by conservation of ADM mass) the mass increase of the black hole is mα (Unruh and Wald 1982). As the mass is lowered arbitrarily close to the horizon, α 0, and so the black hole s mass after the process is carried out, and hence its surface area, will be unchanged but the entropy of the outside world will decrease by s. (This process can even be used to turn heat into work with perfect efficiency, thus violating at least the operational content of the Kelvin statement of the Second Law.) If some large body with mass M and entropy S undergoes gravitational collapse, it will form a black hole with area proportional to M 2, and decrease the entropy of the external world by S. If black hole area is identified with entropy (up to some scale factor K) then the total entropy change is 16πKM 2 S, which for appropriate choices of M and S could easily be negative. (Susskind 1995) As Bekenstein pointed out, both of these arguments would fail if there is some fundamental bound on the minimum size of a body with given entropy and mass. To expand: specialising for simplicity to a Schwarzschild black hole with mass M, the redshift at radial coordinate r is α(r) = (1 2M/r) 1/2 = ((r 2M)/r) 1/2, and the proper distance from the event horizon of an object at coordinate r is d = r 2M dr α(r). (15) Very close to the black hole ((r 2M)/2M 1), we can approximately take ( ) 1/2 r 2M α(r), (16) 2M 17

18 evaluate d, and solve to get α(d) = d/m. (17) So a spherical body of radius d, entropy s, and mass m, lowered slowly into the black hole, will increase the mass of the black hole by δm = md/m, and so the black hole entropy by δs h = 8πMδM = 8πmd. The total increase in (black hole entropy plus outside-matter entropy) is then S = δs h s = 8πmd s. (18) If some new principle of nature means that any such body must satisfy s/m 8πd, that would suffice to ensure S 0 (changing the geometry of the body changes the numerical coefficients but not the overall argument). A qualitatively similar constraint, s/m 2πd, also blocks Susskind s argument from gravitational collapse: the body, on forming a black hole, will have entropy S h = 4πm 2, so the net increase in entropy is S = 4πm 2 s = 2πm(2m s/2πm) 2πm(2m d). (19) But the body must initially lie outside its own Schwarzschild radius, d > 2m, to have avoided collapse already, so this must be positive. However suggestive this Bekenstein bound might be, however, there is at least within classical physics no obvious reason why it must hold. And so to sum up: although classical black holes have some highly thermodynamic-like properties, core aspects of thermodynamics depend on interactions between thermodynamic systems; these interactions do not seem to function correctly for classical black holes, rendering the analogy with thermodynamics purely formal. 4 Quantum field theory Quantum mechanics specifically, quantum field theory, formulated on a classical but curved spacetime removes the blemishes in BHT and transforms it from a suggestive analogy to a full equivalence. The central result here is the Hawking effect: the discovery that black holes emit thermal radiation, at exactly the temperature that BHT would predict. 4.1 Hawking radiation As a starting point to understand the Hawking effect, let s consider for simplicity a free, massless, scalar quantum field theory defined on Schwarzschild spacetime, with metric ds 2 = α(r) 2 dt 2 + α(r) 2 dr 2 + r 2 (dθ 2 + sin 2 (θ)dφ 2 ) (20) where α(r) = 1 2GM/r is the redshift factor. The external region of that spacetime the region outside the event horizon, defined by r > 2GM is a globally hyperbolic spacetime suitable for describing the exterior of an 18